Can you give a definition of the Conway base-13 function better than the one actually present on wikipedia (here), which isn't clear? Maybe with some examples?
You just need to switch back and forth from the lexicographic meaning of the base-13 expansion of the number (think of having ABC instead of .-+) and the loaded meaning you give to the well-formed string as a base-10 number.
An example of a number for which Conway base-13 function is 0 is
where the leftmost . is the threedecimal point (that is, it has a semantic meaning), the three rightmost . mean that the base-13 representation has an infinite number of 1 (that is, they have a metameaning), and the other two . are "digits" of the number (that is, they have a syntactic meaning).
At that number, the function has value -34.11111111111... (in base 10)
The idea of the Conway base-13 number is to find a function that is not continuous, yet if
The function is defined by encoding base-10 values in the tail (the digits left after skipping a finite number). We use
Each number can have up to one base-10 encoded value, which is the result of applying Conway's Base 13 function if it exists. If no such encoding exists for x(ie.
We then show that for each
I understand why the Wikipedia article uses the notation it does, but I find it annoying. Here is a transliteration, with some elaboration.
Note: this function is not computable, as there is no way that you can determine in advance whether the base-13 expansion of x ∈ (0,1) has only finitely many occurances of any of the digits p, m, or d; even if you are provided with a number which is promised to have only finitely many, in general you cannot know when you have found the last one. However, if you are provided with a number x ∈ (0,1) for which you know the location of the final p, m, and d digits, you can compute f(x) very straightforwardly.